Civilization – How does attrition work

In war the AI sometimes gives up and hands over like 12 cities at once for peace (which I'd be stupid not to take)

I think your conclusion here may be false. If those cities are going to drive your happiness into the ground, it may not be in your best interests to take all of them at once. There is no magical wellspring of happiness that will allow you to absorb an unlimited number of enemy cities into your empire without repercussions.

Use the following methods to increase happiness:

• Build courthouses in captured cities. This negates unhappiness from occupation.
• Get as many luxury resources as possible (by building the appropriate improvements on their tiles or through city-state allies). Look for resources just outside your borders and consider purchasing those tiles. Plan new city locations with an eye toward where you can get the most luxury resources.
• Build happiness-producing structures: Circus, Colosseum, Theatre, Stadium, and Burial Tomb if you're Egyptian.
• Build happiness-producing wonders like Hanging Gardens and Eiffel Tower.
• Get social policies (there are many) that work with your existing empire to increase happiness. Look at how they generate happiness, and find the ones that match your style of rule. Don't try to change your whole empire just to eke a little more happiness out of a social policy.

In addition, use puppet cities to your advantage, and don't overextend yourself. If taking all those cities is going to cripple you, it may not be worth it. Is a vast, dysfunctional empire better off than a slightly smaller, functional one?

Sounds like you were unfortunate to not spot the issue earlier; probably the best thing you could have done at that stage would be to fortify your borders, hope to deflect the first wave (when it eventually comes) and then counterattack, charge in and capture and hold a few of their cities (thus making them weaker). Although if you were aiming for a cultural victory you would need to avoid annexing any, otherwise this will make your social policies cost more.

In my experience Montezuma and Augustus are, by default, quite aggressive when played by the AI; if you find an AI playing either near to you, I feel it is best to plan ahead to stop them expanding too much and have sufficient defence on that border - or instead try to make sure you get friendly with them so they're on your side instead.

Also, I believe entering in to a pact of Secrecy with other Civ's against a threatening Civ earlier in the game would have helped prevent the other Civ's becoming allies with the threat later on, by ensuring they have a "bad" relationship with the threat. But I don't have hard evidence for this.